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Unformatted text preview: Department of Mathematical Sciences University of Delaware Prof. T. Angell November 4, 2009 Mathematics 351 Exercise Sheet 6 Exercise 26: (a) Solve the two-point boundary value problem y ￿￿ + 2 y ￿ + y = 0 , y (0) = 2 , y ￿ (2) = −2 . (b) Consider the two-point boundary value problem y ￿￿ + y = 0 , y (0) = 1 , y (π ) = a . For what values of the parameter a will the problem have one solution, many solutions, or no solutions? Exercise 27: For the matrices 1 1 332 0 6 9 5 ,B= A= 2 2 −1 −3 3 0 4 (a) AB , (b) B ￿ A￿ , (c) BC , (d) CA, (e) C −1 . satisfies the equation ￿ ￿ 3 − λ −1 2 A − 3A + 2I = 0. Then compute det (A − λI ) = det . 2 −λ Exercise 29: Find all 2 × 2 matrices X with the property that X 2 = I . Exercise 30: Write down the augmented system matrix for the algebraic system 2 x1 + 3 x2 + 4 x3 x1 + 2 x2 + 3 x3 3 x1 + 5 x2 + 7 x3 = = = 5 4 9 Exercise 28: Show that the matrix A = ￿ 3 −1 2 0 ￿ 1 1 0 and C = 2 0 0 0 0 5 2 0 0 2 0 either find the indicated matrix or explain why it is not defined. 1 0 4 and use elementary row operations to determine the REF of the augmented system matrix. In this way, find all solutions of the original system. Due Date: Wednesday, November 11 in class ...
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